Strange Attractor in Density Evolution

The strange attractor represents a complex pattern of behavior in dynamic systems. This paper introduces a strange attractor of channel polarization as a result of a geometric property of density evolution for code construction. In this way, we can define a subset of synthetic channels that are universally (channel independently) less reliable than the original channel. We show that the cardinality of the attractor set is the n+2-th Fibonacci number for the block length N = 2 n. This can be accepted that it is a significantly large number for very long codes. Recently, it is known that polar codes can be constructed with sub-linear complexity by the use of partial orderings. In this study, we additionally define 1+ log 2 log 2 N universal operators. Finally, we show that these universal operators can be applied on the attractor set to increase the number of synthetic channels that are universally less reliable than the natural channel.